Isospectrality and matrices with concentric circular higher rank numerical ranges Edward Poon,∗ and Hugo J. Woerdeman† Abstract We characterize under what conditions n×n Hermitian matrices A1 and A2 have the property that the spectrum of cos tA1+sin tA2 is independent of t (thus, the trigonometric pencil cos tA1+ n sin tA2 is isospectral). One of the characterizations requires the first ⌈ 2 ⌉ higher rank numerical ranges of the matrix A1 + iA2 to be circular disks with center 0. Finding the unitary similarity between cos tA1 + sin tA2 and, say, A1 involves finding a solution to Lax’s equation. Keywords: Isospectral, trigonometric pencil, higher rank numerical range, Lax pair. AMS subject classifications: 15A22, 15A60 1 Introduction Questions regarding rotational symmetry of the classical numerical range as well as the C−numerical range have been studied in [1, 4, 6, 7, 8]; there is a natural connection with isospectral prop- −it erties. In this paper we study the one parameter pencil Re(e B) = cos tA1 + sin tA2, where 1 ∗ 1 ∗ A1 = ReB = 2 (B + B ) and A2 = 2i (B − B ). We say that the pencil is isospectral when the spectrum σ(Re(eitB)) of Re(eitB) is independent of t ∈ [0, 2π); recall that the spectrum of a square matrix is the set of its eigenvalues, counting algebraic multiplicity. As our main result (Theorem 1.1) shows there is a natural connection between isospectrality and the rotational symmetry of the higher rank numerical ranges of B. Recall that the rank-k numerical range of a square matrix B is defined by Λk(B)= {λ ∈ C : P BP = λP for some rank k orthogonal projection P }. This notion, which generalizes the classical numerical range when k = 1 and is motivated by the arXiv:2103.12885v1 [math.FA] 23 Mar 2021 study of quantum error correction, was introduced in [2]. In [3, 10] it was shown that Λk(B) is convex. Subsequently, in [7] a different proof of convexity was given by showing the equivalence −it −it z ∈ Λk(B) ⇔ Re(e z) ≤ λk(Re(e B)) for all t ∈ [0, 2π). (1) Here λk(A) denotes the kth largest eigenvalue of a Hermitian matrix A. In order to state our main result, we consider words w in two letters. For instance, PPQ, PQPQPP are words in the letters P and Q. The length of a word w is denoted by |w|. When we write na(w, P ) = l we mean that P appears l times in the word w (na=number of appearances). The trace of a square matrix A is denoted by Tr A. ∗Department of Mathematics, Embry–Riddle Aeronautical University, 3700 Willow Creed Road, Prescott, AZ 86301, USA; [email protected] †Department of Mathematics, Drexel University, 3141 Chestnut Street, Philadelphia, PA 19104, USA; [email protected]. 1 Theorem 1.1. Let B ∈ Cn×n. The following are equivalent. (i) The pencil Re(e−itB) = cos t ReB + sin t ImB is isospectral. ∗ k (ii) ∗ Tr w(B,B ) = 0, 0 ≤ l< , 1 ≤ k ≤ n. P|w|=k,na(w,B )=l 2 (iii) For 1 ≤ k ≤ ⌈n/2⌉ the rank-k numerical range of B is a circular disk with center 0, and rank Re(e−itB) is independent of t. (iv) Re(e−itB) is unitarily similar to Re(B) for all t ∈ [0, 2π). Any of the conditions (i)-(iv) imply that B is nilpotent. The paper is organized as follows. In Section 2 we prove our main result. In Section 3 we discuss the connection with Lax pairs. 2 Isospectral paths We will use the following lemma. Lemma 2.1. Let M(t) ∈ Cn×n for t ranging in some domain. Then the spectrum σ(M(t)) is independent of t if and only if TrM(t)k, k = 1,...,n, are independent of t. Proof. The forward direction is trivial. For the other direction, use Newton’s identities to see that the first n moments of the zeros of a degree n monic polynomial uniquely determine the coefficients of the polynomial, and thus the zeros of the polynomial. This implies that TrM(t)k, k = 1,...,n, uniquely determine the eigenvalues of the n × n matrix. Thus, if TrM(t)k, k = 1,...,n, are independent of t, then the spectrum of M(t) is independent of t. k −it k Proof of Theorem 1.1. Consider the trigonometric polynomials fk(t) = 2 Tr[Re(e B)] , k = i(2l−k)t ∗ 1,...,n. The coefficient of e in fk(t) is given by ∗ Tr w(B,B ). By Lemma P|w|=k,na(w,B )=l 2.1 the spectrum of Re(e−itB) is independent of t if and only for k = 1,...,n and 2l 6= k the i(2l−k)t coefficient of e in fk(t) is 0. Due to symmetry, when they are 0 for 2l < k they will be 0 for 2l > k. This gives the equivalence of (i) and (ii). In particular note that when l = 0, we find that TrBk = 0, k = 1,...,n, and thus B is nilpotent. Next, let us prove the equivalence of (i) and (iii). Assuming (i) we have that ReB and −ReB have the same spectrum, so ReB has ⌈n/2⌉ nonnegative eigenvalues. As the spectrum of Re(e−itB) is independent of t, we have that Re(e−itB) has ⌈n/2⌉ nonnegative eigenvalues for all t, guaranteeing −it the rank-k numerical range is nonempty for k ≤ ⌈n/2⌉. Next, since λk(Re(e B)) is independent of t, it immediately follows from the characterization (1) that Λk(B), 1 ≤ k ≤ ⌈n/2⌉, is a circle with center 0. Also, (i) clearly implies that rank(e−itB) is independent of t. Conversely, let us assume (iii). If the rank k-numerical range of B is {z : |z|≤ r} for some r> 0 −it −it −it then λk(Re(e B)) is constant. This also yields that λn+1−k(Re(e B)) = −λk(−Re(e B)). When for 1 ≤ k ≤ ⌈n/2⌉ we have that Λk(B) has a positive radius, we obtain that (i) holds. Next, let us suppose Λℓ(B) has radius zero, and ℓ is the least integer with this property. Then, −it as before, we may conclude that λk(Re(e B)) is a positive constant for 1 ≤ k<ℓ. We also −it −it have, for ℓ ≤ k ≤ ⌈n/2⌉, that λk(Re(e B)) = 0 for some t. As we require rankRe(e B) to −it be independent of t, we find that for ℓ ≤ k ≤ ⌈n/2⌉, λk(Re(e B)) = 0 for all t. Again using −it −it λn+1−k(Re(e B)) = −λk(−Re(e B)), we arrive at (i). The equivalence of (i) and (iv) is obvious. 2 Remark. The condition that rank Re(e−itB) is independent of t in Theorem 1.1(iii) is there to handle the case when Λk(B) has a zero radius. Indeed, it can happen that Λk(B) = {0} without −it λk(Re(e B)) being independent of t; one such example is a diagonal matrix with eigenvalues 1, 0, −1, i. It is unclear whether this can happen for a matrix whose higher rank numerical ranges are disks centered at 0. For sizes 2, 3, and 4, the conditions in Theorem 1.1 are equivalent to B being nilpotent and the numerical range of B being rotationally symmetric. Corollary 2.2. Let B ∈ Cn×n, n ≤ 4. Then the spectrum of Re(e−itB) = cos t ReB + sin t ImB is independent of t if and only if B is nilpotent and the numerical range is a disk centered at 0. Proof. When n = 2, condition (ii) in Theorem 1.1 comes down to TrB = TrB2 = 0. When n = 3 we get the added conditions that TrB3 = TrB2B∗ = 0. When n = 4, we also need to add the conditions TrB4 = TrB3B∗ = 0. The condition that TrBk = 0, 1 ≤ k ≤ n, is equivalent to B being nilpotent. The corollary now easily follows by invoking Remarks 1-3 in [6]. When n = 5, condition Theorem 1.1(ii) says that B is nilpotent and satisfies TrB2B∗ = TrB3B∗ = TrB4B∗ = TrB3B2∗ + TrB2B∗BB∗ = 0. Now to show that Corollary 2.2 does not hold for n ≥ 5, note that the following example from [6], 02000 00000 B = 00011 , 00001 00000 is nilpotent, has the unit disk as its numerical range, but TrB2B∗ = 1 6= 0. 3 Connection with Lax pairs A Lax pair is a pair L(t), P (t) of Hilbert space operator valued functions satisfying Lax’s equation: dL = [P,L], dt where [X,Y ] = XY − Y X. The notion of Lax pairs goes back to [5]. If we start with P (t), and one solves the initial value differential equation d U(t)= P (t)U(t),U(0) = I, (2) dt then L(t) := U(t)L(0)U(t)−1 is a solution to Lax’s equation. Indeed, d L′(t)= [U(t)L(0)U(t)−1]= dt P (t)U(t)L(0)U(t)−1 − U(t)L(0)U(t)−1P (t)U(t)U(t)−1 = P (t)L(t) − L(t)P (t). This now yields that L(t) is isospectral. When P (t) is skew-adjoint, then U(t) is unitary. 3 In our case we have that L(t) = Re(e−itB), and our U(t) will be unitary. This corresponds to P (t) being skew-adjoint. When we are interested in the case when P (t) ≡ K is constant, we have that U(t) = etK .